Question

Square the following general examples to determine the general rule for squaring a binomial:

a. (a + ????)^2
b. (a − ????)^2

Answer 1

See explanation below.

Explanation:

For part a we have the following expression:

`(a+b)^2 = a^2 +2ab + b^2`

And for part b we got:

`(a-b)^2 = a^2 -2ab +b^2`

On general we have the following formula:

`(a+b)^n =\sum_(k=0)^n (nCk) a^(n-k) b^k`

We see that if n=2 we have this:

`(a+b)^2 =\sum_(k=0)^2 (2Ck) a^(2-k) b^k`

`(a+b)^2 = (2C0) a^2 b^0 + (2C1)a^(2-1) b^1 + (2C2) a^(2-2) b^2`

`(a+b)^2 = a^2 +2ab + b^2`

And for the other possibility we have:

`(a+b)^n =\sum_(k=0)^n (-1)^k (nCk) a^(n-k) b^k`

We see that if n=2 we have this:

`(a+b)^2 =\sum_(k=0)^2 (2Ck) a^(2-k) b^k`

`(a+b)^2 = (-1)^0 (2C0) a^2 b^0 +(-1)^1 (2C1)a^(2-1) b^1 + (-1)^2 (2C2) a^(2-2) b^2`

`(a+b)^2 = a^2 +2ab + b^2`

So then we have the general expression for any binomial term elevated at any power.